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ANY-ANGLE PATH PLANNING
by
Alex Nash
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(COMPUTER SCIENCE)
August 2012
Copyright 2012 Alex Nash

Navigating an agent from a given start coordinate to a given goal coordinate through a continuous environment is one of the most important problems faced by roboticists and video game developers. A key part of navigation is path planning. Path planning is the process of generating a path, defined by a sequence of waypoints, that begins at a given start coordinate, ends a given goal coordinate and avoids obstacles in the continuous environment. Path planning is typically composed of two parts: the generate-graph problem, which is solved by discretizing a continuous environment into a graph and the find-path problem, which is solved by searching this graph for a path from a given start vertex to a given goal vertex. Roboticists and video game developers use many different techniques for solving the generate-graph problem, but the find-path problem is typically solved using a traditional edge-constrained find-path algorithm, such as A*. The reason for this is that traditional edge-constrained find-path algorithms have the following desirable properties: Simplicity Property: They are simple to implement and understand. Efficiency Property: They provide a good tradeoff with respect to the runtime of the search and the length of the resulting path. Generality Property: They can be used to search any Euclidean graph. However, these desirable properties come with a penalty. Traditional edge-constrained find-path algorithms propagate information along graph edges and constrain paths to be formed by graph edges. This constraint is artificial and causes the paths found by traditional edge-constrained find-path algorithms to be both longer and less realistic looking than the true shortest paths (that is, the shortest paths in the continuous environment). While this fact is well known by roboticists and video game developers two important questions remain: Question 1: How much longer can the paths found by traditional edge-constrained find-path algorithms be than the true shortest paths? Question 2: Can more sophisticated find-path algorithms be developed that find shorter and more realistic looking paths than traditional edge-constrained find-path algorithms, while maintaining the desirable properties of traditional edge-constrained find-path algorithms? This dissertation addresses these two questions and therefore our hypotheses are as follows: Hypothesis 1: Analytical bounds can be introduced which compare the lengths of the paths found by traditional edge-constrained find-path algorithms on certain types of graphs with the lengths of the true shortest paths. Hypothesis 2: A new class of any-angle find-path algorithms, that propagate information along graph edges, without constraining paths to be formed by graph edges, can be used to quickly find paths that are shorter than the paths found by traditional edge-constrained find-path algorithms, while maintaining the Simplicity and Generality Properties of traditional edge-constrained find-path algorithms. ❧ To validate Hypothesis 1, we introduce a comprehensive set of eight new analytical bounds which compare the lengths of the paths found by traditional edge-constrained find-path algorithms on grid graphs constructed from 2D and 3D regular grids with the lengths of the true shortest paths. ❧ To validate Hypothesis 2, we introduce a new class of any-angle find-path algorithms that propagate information along graph edges (to achieve a short runtime) without constraining paths to be formed by graph edges (to find any-angle paths). We introduce new members to this class and evaluate each member in one of three types of continuous environments, namely continuous 2D environments in which agents have complete knowledge of the environment (known 2D environments), continuous 3D environments in which agents have complete knowledge of the environment (known 3D environments) and continuous 2D environments in which agents do not have complete knowledge of the environment (unknown 2D environments). For each new any-angle find-path algorithm, we use the Simplicity, Efficiency and Generality Properties as our evaluation criteria. Specifically, we introduce three new any-angle find-path algorithms, namely Basic Theta*, Lazy Theta* and Incremental Phi*. In known 2D environments, Basic Theta* satisfies the Simplicity and Generality Properties, provides a good tradeoff, relative to traditional edge-constrained find-path algorithms, with respect to the runtime of the search and the length of the resulting path and a dominating tradeoff over existing any-angle find-path algorithms with respect to the runtime of the search and the length of the resulting path (Efficiency Property). Lazy Theta* is a variant of Basic Theta* that is designed for path planning in known 3D environments. In known 3D environments, Lazy Theta* satisfies the Simplicity and Generality Properties, provides a good tradeoff, relative to traditional edge-constrained find-path algorithms, with respect to the runtime of the search and the length of the resulting path and a dominating tradeoff over Basic Theta* with respect to the runtime of the search and the length of the resulting path (Efficiency Property). Incremental Phi* is a variant of Basic Theta* that is designed for path planning in unknown 2D environments. In unknown 2D environments, Incremental Phi* satisfies the Simplicity Property and provides a dominating tradeoff over Basic Theta* with respect to the runtime of the search and the length of the resulting path (Efficiency Property). These contributions demonstrate that any-angle find-path algorithms represent a promising new technique for path planning in robotics and video games.

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ANY-ANGLE PATH PLANNING
by
Alex Nash
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(COMPUTER SCIENCE)
August 2012
Copyright 2012 Alex Nash